Large language models (LLMs) show remarkable flexibility in adapting to novel tasks without parameter updates, a capacity known as in-context learning (ICL). Prior work has sought to understand ICL by studying the circuits, algorithms, and representations that support it. Yet why some ICL tasks are easy to solve while others are difficult remains unresolved. In this paper, we ask whether LLMs can adapt their representations arbitrarily to solve a simple linear classification task. Specifically, we construct a family of binary classification tasks in which labels are defined by projecting LLMs' own representations onto different axes. Surprisingly, although all tasks are linearly separable by construction, their in-context learnability varies systematically across axes. We find that successful ICL is accompanied by a geometric reorganization of internal representations that increases task-relevant separability. Causal interventions that amplify neural activity along the axis defining the task are insufficient to improve behavioral performance or induce this representational reorganization. We also show that LLM behavior is best described by a prototype-like algorithm operating on representations that are themselves reorganized in context to adapt to the task. Together, these findings offer a geometric account of ICL in LLMs, showing that representations acquired through training constrain what can be exploited through in-context learning.

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